Hilbert's third problem and a conjecture of Goncharov (Q6592194)

Hilbert's third problem asks the following question: given two polyhedra \(P\) and \(Q\), when is it possible to decompose \(P\) into finitely many polyhedra and form \(Q\) out of the pieces? More formally, is it possible to write \(P = \bigcup_{i=1}^n P_i\) and \(Q = \bigcup_{i=1}^n Q_i\) such that \(P_i\cong Q_i\) for all \(i\), and such that \(\text{meas}(P_i\cap P_j ) = \text{meas}(Q_i\cap Q_j) = 0\) for all \(i\not = j\)? (Two polyhedra for which this is true are called \textit{scissors congruent}.) The generalized version of Hilbert's third problem [\textit{J. L. Dupont} and \textit{C.-H. Sah}, J. Pure Appl. Algebra 25, 159--195 (1982; Zbl 0496.52004)] is the observation that this can be asked in any dimension and any geometry. The problem then becomes: describe a complete set of invariants of scissors congruence classes of polytopes in a given dimension and geometry. If two polyhedra are scissors congruent then their volumes are equal. The reverse implication is not true; a second invariant, called the \textit{Dehn invariant}, exists. \N\NThe authors reduce the generalized Hilbert's third problem about Dehn invariants and scissors congruence classes to the injectivity of certain Cheeger-Chern-Simons invariants. Furthermore, the authors establish a version of a conjecture of [\textit{A. Goncharov}, J. Am. Math. Soc. 12, No. 2, 569--618 (1999; Zbl 0919.11080)] relating scissors congruence groups of polytopes and the algebraic K-theory of \(\mathbb{C}\), the complex numbers.